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Low-dimensional lattices. III: Perfect forms. (English) Zbl 0655.10022

An \(n\)-dimensional positive-definite quadratic form \(f\) is perfect if its \(n(n+1)/2\) coefficients are uniquely determined by the system of equations \(f(v^{(i)})=\min (f)\), where \(\pm v^{(1)},\dots,\pm v^{(s)}\) are the minimal vectors. This notion arises in the lattice sphere-packing problem (extreme forms are perfect). As the authors say, “progress in the study of perfect lattices has been steady but slow”. After the problem for \(n\leq 5\) had been solved in the last century, the enumeration for \(n=6\) was completed by E. S. Barnes [Acta Arith. 5, 57–79 and 205–222 (1959; Zbl 0083.04201 and Zbl 0089.02706)], but the case \(n=7\) is still not fully solved.
The first part of the paper contains an elegant proof that the \(A_ 1\), \(A_ 2\), \(A_ 3\), \(A_ 4\) and \(D_ 4\) root lattices give the only perfect forms for \(n\leq 4\). If \(L\) is a perfect lattice, the obvious condition \(s\geq n(n+1)/2\) and general properties of minimal vectors under congruence mod 2 immediately lead to a sublattice of \(L\) which is the desired root lattice; knowledge of its covering radius then shows that this must be all of \(L\). In this way the authors give also an alternative approach to the maximum density of a lattice sphere-packing for \(n\leq 4\) (modulo the general fact that a maximum is attained).
The main part of the paper describes the perfect forms for \(n=5,6\) and, above all, the 33 known perfect forms for \(n=7\) [cf. K. C. Stacey, J. Lond. Math. Soc. (2) 10, 97–104 (1975; Zbl 0297.10012) and J. Aust. Math. Soc., Ser. A 22, 144–164 (1976; Zbl 0332.10014)]. This is done in a more systematic and complete way than in the existing literature. In particular, 30 of those 33 forms are found to be extreme. The authors have made their calculations as a step towards the definite enumeration of the perfect seven-dimensional forms which is expected to confirm the known list.
Reviewer: H.G.Quebbemann

MSC:

11E16 General binary quadratic forms
11H55 Quadratic forms (reduction theory, extreme forms, etc.)
11H06 Lattices and convex bodies (number-theoretic aspects)
11H31 Lattice packing and covering (number-theoretic aspects)
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